Stream function

So far in mapping out potential flow directions we have merely noted that the streamline at any point must be perpendicular to the equipotential line, and we have sketched such streamlines. Now we introduce a more formal method of showing the flow directions at any point. From Eq. 3.33 we know that for a steady, incompressible, two-dimensional flow 

We now arbitrarily define a function if/, called the stream function, or Lagrange stream function, by the equations 

So any flow that satisfies this equation automatically satisfies the two-dimensional, steady-flow, incompressible-material balance. From this equation we see that iff must also have the dimension of square feet or meters per second. Comparing Eqs.jlO.7 and 10,39, we find 

It is proved in i pp. H that Eqs. 10.41 and 10.42 can be satisfied only if for every x and y the curve of constant p and the curve of constant ip passing through that point are perpendicular. This is illustrated by Fig, 10.13 (see also Prob. 10.7). I 

Since streamlines also have the property of being everywhere perpendicular to equipoteiitials, all the streamlines which we have drawn in Figs. 10.3, 10.4, 10.5, 10.7, 10,8, and 10.10 are lines of constant ip. 

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Equation 9 is Laplace s equation which also occurs in several other fields of mathematical physics. Where the flow problem is two-dimensional, the velocities ate also detivable from a stream function, /. 

The Stream Function Stream functions are defined for two-dimensional flow and for three-dimensional axial symmetric flow. The stream function can be used to plot the streamlines of the flow and find the velocity. For two-dimensional Bow the velocity components can be calculated in Cartesian coordinates by... 

Superposition of Flows Potential flow solutions are also useful to illustrate the effect of cross-drafts on the efficiency of local exhaust hoods. In this way, an idealized uniform velocity field is superpositioned on the flow field of the exhaust opening. This is possible because Laplace s equation is a linear homogeneous differential equation. If a flow field is known to be the sum of two separate flow fields, one can combine the harmonic functions for each to describe the combined flow field. Therefore, if d)) and are each solutions to Laplace s equation, A2, where A and B are constants, is also a solution. For a two-dimensional or axisymmetric three-dimensional flow, the flow field can also be expressed in terms of the stream function. 

Flow Past a Point Sink A simple potential flow model for an unflanged or flanged exhaust hood in a uniform airflow can be obtained by combining the velocity fields of a point sink with a uniform flow. The resulting flow is an axially symmetric flow, where the resulting velocity components are obtained by adding the velocities of a point sink and a uniform flow. The stream function for this axisymmetric flow is, in spherical coordinates. 

Stream function has value P = q/ikv) at the dividing streamline and so its location can be found by expressing Eq. (10.41) in polar coordinates ... 

Outside the jet and away from the boundaries of the workbench the flow will behave as if it is inviscid and hence potential flow is appropriate. Further, in the central region of the workbench we expect the airflow to be approximately two-dimensional, which has been confirmed by the above experimental investigations. In practice it is expected that the worker will be releasing contaminant in this region and hence the assumption of two-dimensional flow" appears to be sound. Under these assumptions the nondimensional stream function F satisfies Laplace s equation, i.e.. 

In the case of the free jet, the solution for the Aaberg exhaust system can be found by solving the Laplace equation by the method of separation of variables and assuming that there is no fluid flow through the surface of the workbench. At the edge of the jet, which is assumed to be at 0—0, the stream function is given by Eq. (10.113). This gives rise to... 

Following the procedure discussed previously, we find that at large distances from the Aaberg exhausr hood the stream function is given by... 

Therefore, a stream function T may be introduced in the meridian plane of the cyclone, i.e., the r, 9) plane in the spherical coordinate system ... 

The streamlines calculated for different values of the stream function are shown in Fig. 32. Some flow recirculation is visible near the corners, but this is negligible as compared to the viscoelastic situation [110]. 

Fig. 32. Streamlines in abrupt contraction flow computed at a midtravel distance of the piston at x = — 40 mm (the origin of the abscisse is taken at the orifice entrance). Due to axial symmetry, only half of the flow tube is shown. The dimensionless stream function

Figure 6.12. Particle stream functions, ty(e = 0.65) (Radial position is expressed as fraction of radial distance from centre line, and axial position as fraction of bed height measured from the bottom)(43 ...

The stream function satisfying the fourth-order differential equation, used by Haberman and Sayre (H2) is... 

If the equations of motion, the continuity relationship, and the proper stream function are properly combined, equations result which enable one to plot the flow lines for both internal and external motion. The stream function for an infinite extent of continuous phase around a single drop is (B7, H2)... 

From the definition of a particle used in this book, it follows that the motion of the surrounding continuous phase is inherently three-dimensional. An important class of particle flows possesses axial symmetry. For axisymmetric flows of incompressible fluids, we define a stream function, ij/, called Stokes s stream function. The value of Imj/ at any point is the volumetric flow rate of fluid crossing any continuous surface whose outer boundary is a circle centered on the axis of symmetry and passing through the point in question. Clearly ij/ = 0 on the axis of symmetry. Stream surfaces are surfaces of constant ij/ and are parallel to the velocity vector, u, at every point. The intersection of a stream surface with a plane containing the axis of symmetry may be referred to as a streamline. The velocity components, and Uq, are related to ij/ in spherical-polar coordinates by... 

It is often convenient to work in terms of a dimensionless stream function and vorticity defined, respectively, as... 

The system considered in this chapter is a rigid or fluid spherical particle of radius a moving relative to a fluid of infinite extent with a steady velocity U. The Reynolds number is sufficiently low that there is no wake at the rear of the particle. Since the flow is axisymmetric, it is convenient to work in terms of the Stokes stream function ij/ (see Chapter 1). The starting point for the discussion is the creeping flow approximation, which leads to Eq. (1-36). It was noted in Chapter 1 that Eq. (1-36) implies that the flow field is reversible, so that the flow field around a particle with fore-and-aft symmetry is also symmetric. Extensions to the creeping flow solutions which lack fore-and-aft symmetry are considered in Sections II, E and F. 

Corresponding streamlines are shown in Figs. 3.3b and 3.4b. Like the creeping flow result, the Oseen solution predicts infinite drift. For large r the velocity is unbounded, but the divergent terms are 0[Re ] and formally beyond the range of the Oseen approximation. For r <2/Re, the stream function may be approximated as... 

Gupalo and Ryazantsev (GIO) followed the analysis of Acrivos and Taylor (A2) with the Proudman-Pearson stream function rather than Stokes flow. For Sc > 10, the two predictions for Sh agree within 1%, while for Sc = 1 they differ by at most 8% for Pe < 1. The results of Gupalo and Ryazantsev, although valid to higher Re, are still restricted to Pe 0, so that this extension is of little practical value. 

A useful theorem due to Payne and Pell (P3) enables the drag on an axisymmetric body to be calculated directly from the stream function ij/ for steady... 

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